# Second Prairie Analysis Seminar

## Location and Dates:

### Department of Mathematics

University of Kansas

Lawrence, Kansas

October 18-19, 2002

#### Principal Lecturer:

### Daniel Tataru

#### University of California, Berkeley

#### Invited Speakers:

### Camil Muscalu

#### University of California, Los Angeles

### Wilhelm Schlag

#### California Institute of Technology

#### Organizers:

Estela A. Gavosto, KU

Marianne Korten, KSU

Charles Moore, KSU

Rodolfo H. Torres, KU

#### Contact Information:

If you would like further information, please contact **torres@math.ku.edu**

## Tentative schedule

All the talks will be given in Room 120 Snow Hall.

Friday October 18 |
||

TIME | SPEAKER | TALK TITLE |

1:00-1:25 | Registration | |

1:25-1:30 | Welcome remarks | |

1:30-2:30 |
Daniel Tataru UC Berkeley |
Nonlinear wave equations and wave parametrices |

2:30-2:35 | Short break | |

2:35-3:35 |
Camil Muscalu UCLA |
Multilinear singular integrals |

3:35-4:00 | Tea time | |

4:00-4:25 |
Nikolaos Tzirakis U. Massachusetts |
Global existence for the semilinear quintic NLS |

4:25-4:50 |
Michael Scott Kansas State U. |
An exact, lightlike, shock-wave solution of the Einstein equations. |

4:50-5:15 |
Caroline Sweezy New Mexico State U. |
Weighted inequalities and Littlewood-Paley functions for parabolic solutions on non-smooth domains |

8:00 - ... | After-dinner party at Estela's and Rodolfo's home |

Saturday October 19 |
||

TIME | SPEAKER | TALK TITLE |

8:55-9:20 |
Pietro Poggi-Corradin Kansas State U. |
Harmonic functions on homogeneous domains |

9:20-9:45 |
Alexander Stokolos U. Connecticut |
Tangential boundary behavior of bounded harmonic functions in the unit disc |

9:45-10:00 | Coffee break | |

10:00-10:25 |
Dimitry Ryabogin U. Missouri |
Volumes of projections of convex bodies via Fourier transform |

10:25-10:50 |
Arpad Benyi U. Massachusetts |
Symbolic calculus for bilinear pseudodifferential operators |

10:50-11:00 | Coffee break | |

11:00-12:00 |
Wilhelm Schlag Caltech |
On the Hardy-Littlewood majorant problem for random sets |

12:00 - 1:20 | Lunch provided by the Department of Mathematics | |

1:30-2:30 |
Daniel Tataru UC Berkeley |
Sharp local well-posedness for nonlinear wave equations |

2:30-2:40 | Coffee break | |

2:40-3:05 |
Robert Smits New Mexico State U. |
Conformal mappings and Brownian motion |

3:05-3:30 |
Georgiy Arutyunyants U. Missouri |
About singular approximation to the identity |

3:30-... | Tea time, informal discussions, adjournment. |

## Talks and Abstracts

### Daniel Tataru (slides of the talks)

**1. Nonlinear wave equations and wave parametrices**

The aim of this talk is to provide an introduction for the local theory for nonlinear wave equation. Also we describe some of the tools for the parametrix constructions for linear wave equations, and their use in the study of nonlinear waves.

**2. Sharp local well-posedness for nonlinear wave equations**

The purpose of this talk is to describe the recent joint work with Hart Smith, whose aim is to establish sharp local results for nonlinear wave equations.

### Camil Muscalu

**Multilinear singular integrals**

We will review some results of the "modern" theory of multilinear singular integrals (and their maximal "Carleson type" analogues), as they have been developed over the last few years. Most of the theorems we are going to present, have been obtained in collaboration with Terence Tao and Christoph Thiele.

### Wilhelm Schlag (slides of the talk)

**On the Hardy-Littlewood majorant problem for random sets**

Hardy and Littlewood observed that L^p norms of functions on the circle obey a certain monotonicity property with respect to the Fourier coefficients provided that p is an even integer. If p>2 is not an even integer, then it is known that this property will fail. The question arises whether it remains correct for trigonometric polynomials in some quantitative form depending on the degree. We will present recent Work with Gerd Mockenhaupt on this question.

## Contributed Talks

Georgiy Arutyunyants

Title: About singular approximation to the identity

Abstract: We will discuss some geometrical techniques to attack a weak-type problem for a maximal operator with a rough kernel

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Arpad Benyi

Title: Symbolic calculus for bilinear pseudodifferential operators

Abstract: We introduce several natural classes of bilinear pseudodifferential operators. We study their boundedness properties and discuss a symbolic calculus for the transposes of certain operators of order zero.

*****************************************

Pietro Poggi-Corradini

Title: Harmonic functions on homogeneous domains

Abstract: We describe recent advances in the description of the cone of positive harmonic functions vanishing on the finite boundary of certain planar domains; its connections to the study of entire functions and to solutions of a certain partial differential equation on subdomains of the torus; and we mention an interesting open problem.

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Dmitry Ryabogin

Title: Volumes of projections of convex bodies via Fourier transform.

Abstract: In this talk we present the Fourier analytic approach to projections of convex bodies based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function. (This is a joint work with A. Koldobsky and A. Zvavitch)

*****************************************

Michael Scott

Title: An Exact, Lightlike, Shock-Wave Solution of the Einstein Equations.

Abstract: Using an extension of a shock-matching theory first developed by Joel Smoller and Blake Temple we construct a new exact solution of the Einstein equations, which can be interpreted as an outgoing spherical shock wave that propagates at the speed of light. The solution is constructed by matching a Friedman Robertson Walker (FRW) metric, which is a geometric model for the universe, to a Tolman Oppenheimer Volkoff (TOV) metric, which models a static isothermal spacetime. The pressure and density are finite on each side of the shock throughout the solution, and the sound speeds, on each side of the shock, are constant and sub-luminous. Moreover, the pressure and density are smaller at the leading edge of the shock, which is consistent with the Lax entropy condition in classical gas dynamics. However, the shock speed is greater than all the characteristic speeds. The solution also yields a surprising result in that the solution is not equal to the limit of previously known sub-luminous solutions as they tend to the speed of light.

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Robert Smits

Title: Conformal Mappings and Brownian Motion

Abstract: Much work has been done recently on heat kernels and Brownian Motion in unbounded domains especially related to domains of finite area, conical domains or convex domains of parabolic type. In this talk I would like to discuss recent work which shows how in two dimensions conformal mappings can be used to extend these results to more general domains showing that the essence of the estimates depend on the growth of the inradius along quasigeodesics.

*****************************************

Alex Stokolos

Title: Tangential boundary behavior of bounded harmonic functions in the unit disc

Abstract: Bounded harmonic functions in the unit disc $D$ converge nontangentially almost everywhere (Fatou, 1906) and fail to converge along the rotates of any given tangential curve (Littlewood, 1927). We study their boundary behaviour along tangential curves whose shape may change from point to point (a problem posed by W. Rudin). Let $\tau$ be the assignment of a curve $\tau_\theta$ in $D$ ending at $\theta$ and tangential to the boundary $bD$ of $D$, for each $\theta\in bD$. The authors announce a proof that convergence along $\tau$ fails if $\tau_\theta$ depends on $\theta$ in a measurable way, and to show that there is a family $\tau$ of tangential curves such that each bounded harmonic function in $D$ converges along $\tau_\theta$ for a set of points $\theta$ whose outer measure is equal to $2\pi$.

*****************************************

Caroline Sweezy

Titile: Weighted inequalities and Littlewood-Paley functions for parabolic solutions on non-smooth domains

Abstract: Sufficient conditions for weighted inequalities involving parabolic gradients and their boundary functions on non-smooth domains are established by first proving a discrete Littlewood-Paley inequality. This is joint work wih J. Michael Wilson.

******************************************

Nikos Tzirakis

Title: Global existence for the semilinear quintic NLS

Abstract: In this talk I'll show how,using the I-method of J.Colliander,M.Keel,G.Staffilani,H.Takaoka and T.Tao, we can get global well posednesss results for the quintic defocusing NLS on R with initial data that are below the energy norm.

#### Organizers:

Estela A. Gavosto, KU

Marianne Korten, KSU

Charles Moore, KSU

Rodolfo H. Torres, KU